The quantum Swiss roll or turf in space

Here are some idle thoughts concerning mostly inertia.
Really I'm curious to know what happens and then what variations
or principles can be derived.

Imagine a planar piece of material floating in space.
In our frame of reference it is completely stationary.
Now by some internal magic one edge of the material starts
to curl up as if it wanted to become a Swiss roll.
(Fig 1)
When it is finished curling:-

Is the roll rotating?

Where is the centre of gravity and is it stationary?

What difference does it make if we curl quickly or slowly?

Well how about if now the roll unrolls all together instead
of from one end? (Fig 2) One type of this might happen when
a rotating ring breaks. (Fig 5) When it is finished straightening
itself out does it return to the original state?
Either way I want to know for certain.

There are lots of subsidiary questions such as does the speed matter,
what is the effect of a travelling wave. What difference does the
elasticity or natural tendency to stay in shape have?
This could be a quantum thing.

I'm also keen to see exactly what work is done by the system
during a curl. Imagine for the sake of argument a set of hinges
with a hydraulic actuator. To effect a curl, force-times-distance
is used to start the massive elements on either side of the hinge moving.
Then some decelerating force is required to stop the hinge rotating forever.
It is easy to imagine a simple, perfect, system where the energy spent on
accelerating the links is completely recovered when stopping the movement;
but can we be sure this is the case here?

There are plenty of alternative curling and uncurling possibilities.
Figs 3 and 4 show some plane configurations but there are more if we
consider helical wire such as would occur if our rectangular plane was
replaced by a Vee which started curling from the bottom... and so on.

A curl (which of course could be a travelling wave) takes time to traverse
the width of the plane, are there cases, perhaps in quantum-land, where there
isn't enough time and so the curl has to happen at once? (Fig 5) Now you can
see why I'm not really looking at inter-stellar drives but the principles.
Figure 5 could be a stable 3-configuration being opened-up ready to become
a 4-configuration with the addition of another link... ...But who knows if
the new configuration is stable? Not a question we can answer with such a
model but we might find it a useful analogue for quantum events.

How to find out? Would somebody volunteer to write a model that looks something
like a bicycle chain. My mechanics is just not reliable enough to make sure
I've taken everything into account.

If it all sounds rather boring, think of what
happens to the tail-end-charlie of a conga when it goes round corners - Some whip!